Interpretation of Atomic Structure Factor Curves in Crystal Reflection of X-Rays

In a previous paper (<article>Phys. Rev., May, 1928</article>) the authors showed that, for certain values of D the grating space of a crystal of rocksalt, the area under a radial electron distribution (or U) curve for chlorine rose above 19 electrons. This result was obtained both from Havighurst's experimental F curve and from F values calculated for a model chlorine ion, these calculated values being modified to take into account the Compton effect. This result seemed perplexing, inasmuch as both the real ion and the model ion have but 18 electrons. The present paper is a further discussion of this point. It is proved that for any symmetrical atom F values, calculated according to the classical theory and unmodified for the Compton effect but multiplied by the Debye temperature factor, give U curves the areas under which never exceed the number of electrons assumed in the model. It is also shown that an unsymmetrical atom gives F values which behave in the same way. But, since both experimental and modified theoretical F values (that is, modified to take account of the Compton effect) give U curves the areas under which do exceed the true number of electrons for certain values of D, there is an indication that the Compton effect is involved in the experimental values. The truth of this indication would invalidate the use of the Fourier analysis method as now applied. The present paper also develops the Fourier integral as a quick method of calculating a U curve from a model atom on the classical theory. It is shown that a U curve calculated from Compton's formula <math display="display">U=(8πr/D)Σ<under>1</under>∞<operand>(nF_n/D)<sin></sin>(2πrn/D)</operand> for a Fourier series is a very close approximation to the true U curve given by the Fourier integral <math display="display">U(r)=8πr<int>₀^∞<operand>xF(λx/2)<sin></sin>2πrx dx</operand></int> where F is the same function of (λx/2) as F in the series formula is a function of <sin></sin>θ. An analysis by the Fourier integral of a model supposed to have all the electrons concentrated at the center together with the Debye temperature factor shows that U(r) represents the distribution of electrons about a lattice point and not about the center of the atom.